If you are trying to copy a "block of files" (multiple files), and the transfer stops, it could be that one (or more) of the files is corrupted or otherwise exhibits a problem to the OS. The result is that the copy just "quits at the bad file", leaving the rest "UN-copied".

What I would do:
Get a paper and pencil to keep written notes.
Reduce the size of your copying, down to single folders or files if need be.
The object here is to slowly weed out the bad or un-copyable file(s) from the good ones.

Could this be a problem with file permissions?

I'd suggest you open a google search page and enter "mac file error 32" into the search box and start reading up...

MacRumors attracts a broad audience
of both consumers and professionals interested in
the latest technologies and products. We also boast an active community focused on
purchasing decisions and technical aspects of the iPhone, iPod, iPad, and Mac platforms.